What is the difference between all those types?

Special features:
gta02 - accurate and sophisticated reporting of capacity,
time_to_full, time_to_empty, temperature and battery current during
both charge and discharge thanks to bq27000 (aka Coloumb Counter)

What are hardware capabilities of gta01 and gta02 with regard to battery management?

gta01: charging all battery types, measuring temperature with
battery-integrated thermistor (currently charging and measuring
temperature for non-gta01 batteries doesn't work due to the kernel
driver issues but it's software limitation), measuring battery output
voltage, very inaccurate and noisy measuring of battery current

Can nokia phones use/charge gta01/gta02 batteries?

gta01 and gta02 batteries will fit wherever BL-6C fits but they
can't be charged in nokia phones unless you isolate the middle pin
from the battery and connect a resistor of ~50k (actual measured value
on a cold (25C) battery is 75k, on a slightly warm battery - 82k) from
it to the ground (to fake a thermistor presence).

Can third-party chargers charge gta01/gta02 batteries?

The "good" ones will most probably require the same trick needed
for nokia phones. More cheaper ones are more likely to ignore
thermistor absence.

I have several compatible batteries. What are the storage requirements for them?

Keep in a dry cool place charged to no more than 75%.

Safety issues

Do OM devices control temperature to stop charging if the battery gets too hot?

No (probably gta01 does, need to check).

Isn't it dangerous?

No, since all batteries (not raw cells!) have an integrated
protection circuits.

Can i use that fancy 2800 mAh BL-5C-compatible battery i saw on ebay?

Unless you want an explosion in your pocket i wouldn't recommend
using any battery that is not produced by a reputable vendor and
widely tested. And even reputable vendors make mistakes, nokia once
had to recall 46 million batteries manufactured by Matsushita ([1]).

You say that BL-5C is compatible with my gta02. Does that mean i can use that BL-5C-compatible bat i bought for a buck from a bum?

Charging

LiIon batteries don't like to be kept fully charged, so the charger
stops as soon as charging current becomes less than threshold. If you
have GSM on it will discharge the battery.

But why doesn't it ever stop charging on my device?

The GSM modem is connected directly to the battery terminals so if
it's active, charger will think it's still charging the battery and
won't turn off unless GSM becomes inactive. The default threshold is
about 16mA, the latest Qi increases the threshold to ~32mA.

=== Does it mean if i leave my phone plugged it will eventually fully
discharge the battery? ===

On gta02 the charger will restart the charge automatically once the
battery voltage reaches ~4V which corresponds to ~76%. Not sure about
gta01, requires more investigation. :-/

=== Ok, how to make sure my battery is fully charged before a long
trip? ===

Replug the charger, it will trigger charging no matter what the
current capacity is.

=== My power/aux LED indicates charging/discharging/whatever, what does
that mean (aka why it's still blue even after i unplugged the
charger)? ===

Ask FSO guys about it, some of them think that the user shouldn't
really know what's happening and therefore they do some special
mangling of "status" sysfs node before presenting it to the user. If
you want to make a decent bugreport please add clear steps to
reproduce and /sys/class/power_supply/battery/uevent contents for all
relevant states.

What is the difference between all those types?

Special features:
gta02 - accurate and sophisticated reporting of capacity,
time_to_full, time_to_empty, temperature and battery current during
both charge and discharge thanks to bq27000 (aka Coloumb Counter)

What are hardware capabilities of gta01 and gta02 with regard to battery management?

gta01: charging all battery types, measuring temperature with
battery-integrated thermistor (currently charging and measuring
temperature for non-gta01 batteries doesn't work due to the kernel
driver issues but it's software limitation), measuring battery output
voltage, very inaccurate and noisy measuring of battery current

Can nokia phones use/charge gta01/gta02 batteries?

gta01 and gta02 batteries will fit wherever BL-6C fits but they
can't be charged in nokia phones unless you isolate the middle pin
from the battery and connect a resistor of ~50k (actual measured value
on a cold (25C) battery is 75k, on a slightly warm battery - 82k) from
it to the ground (to fake a thermistor presence).

Can third-party chargers charge gta01/gta02 batteries?

The "good" ones will most probably require the same trick needed
for nokia phones. More cheaper ones are more likely to ignore
thermistor absence.

I have several compatible batteries. What are the storage requirements for them?

Keep in a dry cool place charged to no more than 75%.

Safety issues

=== Do OM devices control temperature to stop charging if the battery
gets too hot? ===

No (probably gta01 does, need to check).

Isn't it dangerous?

No, since all batteries (not raw cells!) have an integrated
protection circuits.

=== Can i use that fancy 2800 mAh BL-5C-compatible battery i saw on
ebay? ===

Unless you want an explosion in your pocket i wouldn't recommend
using any battery that is not produced by a reputable vendor and
widely tested. And even reputable vendors make mistakes, nokia once
had to recall 46 million batteries manufactured by Matsushita ([1]).

=== You say that BL-5C is compatible with my gta02. Does that mean i
can use that BL-5C-compatible bat i bought for a buck from a bum? ===

Charging

LiIon batteries don't like to be kept fully charged, so the charger
stops as soon as charging current becomes less than threshold. If you
have GSM on it will discharge the battery.

But why doesn't it ever stop charging on my device?

The GSM modem is connected directly to the battery terminals so if
it's active, charger will think it's still charging the battery and
won't turn off unless GSM becomes inactive. The default threshold is
about 16mA, the latest Qi increases the threshold to ~32mA.

=== Does it mean if i leave my phone plugged it will eventually fully
discharge the battery? ===

On gta02 the charger will restart the charge automatically once the
battery voltage reaches ~4V which corresponds to ~76%. Not sure about
gta01, requires more investigation. :-/

=== Ok, how to make sure my battery is fully charged before a long
trip? ===

Replug the charger, it will trigger charging no matter what the
current capacity is.

=== My power/aux LED indicates charging/discharging/whatever, what does
that mean (aka why it's still blue even after i unplugged the
charger)? ===

Ask FSO guys about it, some of them think that the user shouldn't
really know what's happening and therefore they do some special
mangling of "status" sysfs node before presenting it to the user. If
you want to make a decent bugreport please add clear steps to
reproduce and /sys/class/power_supply/battery/uevent contents for all
relevant states.